Optimal. Leaf size=147 \[ -\frac {\sinh ^4(x) (b-a \coth (x))}{4 \left (a^2-b^2\right )}-\frac {\sinh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \coth (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac {a^4 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^3}-\frac {a (3 a+b) \log (1-\coth (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (\coth (x)+1)}{16 (a-b)^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.34, antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3516, 1647, 801} \[ -\frac {a^4 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^3}-\frac {\sinh ^4(x) (b-a \coth (x))}{4 \left (a^2-b^2\right )}-\frac {\sinh ^2(x) \left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \coth (x)\right )}{8 \left (a^2-b^2\right )^2}-\frac {a (3 a+b) \log (1-\coth (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (\coth (x)+1)}{16 (a-b)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 801
Rule 1647
Rule 3516
Rubi steps
\begin {align*} \int \frac {\cosh ^4(x)}{a+b \coth (x)} \, dx &=-\left (b \operatorname {Subst}\left (\int \frac {x^4}{(a+x) \left (-b^2+x^2\right )^3} \, dx,x,b \coth (x)\right )\right )\\ &=-\frac {(b-a \coth (x)) \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {a^2 b^4}{a^2-b^2}-\frac {3 a b^4 x}{a^2-b^2}+4 b^2 x^2}{(a+x) \left (-b^2+x^2\right )^2} \, dx,x,b \coth (x)\right )}{4 b}\\ &=-\frac {\left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \coth (x)\right ) \sinh ^2(x)}{8 \left (a^2-b^2\right )^2}-\frac {(b-a \coth (x)) \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {\operatorname {Subst}\left (\int \frac {\frac {a^2 b^4 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^2}-\frac {a b^4 \left (5 a^2-b^2\right ) x}{\left (a^2-b^2\right )^2}}{(a+x) \left (-b^2+x^2\right )} \, dx,x,b \coth (x)\right )}{8 b^3}\\ &=-\frac {\left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \coth (x)\right ) \sinh ^2(x)}{8 \left (a^2-b^2\right )^2}-\frac {(b-a \coth (x)) \sinh ^4(x)}{4 \left (a^2-b^2\right )}-\frac {\operatorname {Subst}\left (\int \left (-\frac {a b^3 (3 a+b)}{2 (a+b)^3 (b-x)}+\frac {8 a^4 b^4}{(a-b)^3 (a+b)^3 (a+x)}-\frac {a (3 a-b) b^3}{2 (a-b)^3 (b+x)}\right ) \, dx,x,b \coth (x)\right )}{8 b^3}\\ &=-\frac {a (3 a+b) \log (1-\coth (x))}{16 (a+b)^3}+\frac {a (3 a-b) \log (1+\coth (x))}{16 (a-b)^3}-\frac {a^4 b \log (a+b \coth (x))}{\left (a^2-b^2\right )^3}-\frac {\left (4 b \left (2 a^2-b^2\right )-a \left (5 a^2-b^2\right ) \coth (x)\right ) \sinh ^2(x)}{8 \left (a^2-b^2\right )^2}-\frac {(b-a \coth (x)) \sinh ^4(x)}{4 \left (a^2-b^2\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.57, size = 144, normalized size = 0.98 \[ \frac {12 a^5 x+a^5 \sinh (4 x)-32 a^4 b \log (a \sinh (x)+b \cosh (x))+24 a^3 b^2 x-2 a^3 b^2 \sinh (4 x)-b \left (a^2-b^2\right )^2 \cosh (4 x)-4 b \left (3 a^4-4 a^2 b^2+b^4\right ) \cosh (2 x)+8 a^3 \left (a^2-b^2\right ) \sinh (2 x)-4 a b^4 x+a b^4 \sinh (4 x)}{32 (a-b)^3 (a+b)^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.43, size = 1229, normalized size = 8.36 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 216, normalized size = 1.47 \[ -\frac {a^{4} b \log \left ({\left | -a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {{\left (3 \, a^{2} - a b\right )} x}{8 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, x\right )} - 6 \, a b e^{\left (4 \, x\right )} + 8 \, a^{2} e^{\left (2 \, x\right )} - 12 \, a b e^{\left (2 \, x\right )} + 4 \, b^{2} e^{\left (2 \, x\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 8 \, a e^{\left (2 \, x\right )} + 4 \, b e^{\left (2 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.13, size = 319, normalized size = 2.17 \[ -\frac {a^{4} b \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}+\frac {1}{\left (4 a +4 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {4}{\left (8 a +8 b \right ) \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {7 a}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {5 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {5 a}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) a^{2}}{8 \left (a +b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right ) a b}{8 \left (a +b \right )^{3}}-\frac {1}{\left (4 a -4 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {4}{\left (8 a -8 b \right ) \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {5 a}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {3 b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {7 a}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {5 b}{8 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) a^{2}}{8 \left (a -b \right )^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right ) a b}{8 \left (a -b \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.34, size = 154, normalized size = 1.05 \[ -\frac {a^{4} b \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} + a + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {{\left (3 \, a^{2} + a b\right )} x}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac {{\left (4 \, {\left (2 \, a + b\right )} e^{\left (-2 \, x\right )} + a + b\right )} e^{\left (4 \, x\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} - \frac {4 \, {\left (2 \, a - b\right )} e^{\left (-2 \, x\right )} + {\left (a - b\right )} e^{\left (-4 \, x\right )}}{64 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.74, size = 135, normalized size = 0.92 \[ \frac {{\mathrm {e}}^{4\,x}}{64\,a+64\,b}-\frac {{\mathrm {e}}^{-4\,x}}{64\,a-64\,b}-\frac {{\mathrm {e}}^{-2\,x}\,\left (2\,a-b\right )}{16\,{\left (a-b\right )}^2}+\frac {{\mathrm {e}}^{2\,x}\,\left (2\,a+b\right )}{16\,{\left (a+b\right )}^2}-\frac {a^4\,b\,\ln \left (b-a+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {a\,x\,\left (3\,a-b\right )}{8\,{\left (a-b\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{4}{\relax (x )}}{a + b \coth {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________